Key Highlights Tencent Hunyuan has announced that, working through the research agent Hyra together with the Hy3 model, its team constructed an integer set A for which the exponential ratio between |A+A| and |A-A| reaches exactly 2. That settles the extremal problem of sum-difference sets, a question left open since 1969. Across more than fifty years before this, the best constructions mathematicians produced pushed that exponent only slightly past 1.1. The new work does not merely clear that barrier, it also proves that 2 is the optimal exponent for the problem, which means this particular road has now been walked all the way to its end. Both the paper and a formal proof have been made public, so the result is open to inspection rather than sitting behind a press release. For a question that had stood for half a century, that is an unusually clean ending. ## How It Unfolded The starting point traces back to 1969. Mathematicians began asking how large a gap can open up, in terms of size, between the sumset A+A and the difference set A-A for a finite set of integers A. Intuition says the difference set should be the larger of the two, because addition produces duplicates through commutativity while subtraction does not. Exactly how far that intuition could be overturned, however, never had a definitive answer. For the half century that followed, progress along this direction was decidedly slow. Researchers kept building ever more intricate sets, nudging the exponent upward bit by bit, yet the field stayed stuck at a level only slightly above 1.1. On a problem that old, incremental gains had become the norm, and nobody was expecting the ceiling itself to be reached outright. The approach this time was to plug the research agent Hyra into the Hy3 model and let the system search, verify, and iterate autonomously across an enormous construction space. What came out was an explicit construction achieving an exponent of 2, paired with an argument establishing that this value is optimal. ## Technical Details One point deserves to be spelled out: Hyra is not playing the role of a coding assistant here, it functions as a research agent. It has to propose candidate constructions on its own, judge which paths are worth pursuing in depth, adjust strategy after failures, and then organize its results into a form that can be checked. That is fundamentally different from conversational question answering, because the search space in mathematical construction problems is enormous and the overwhelming majority of attempts end in failure. Whether a system can prune efficiently is the whole game. Raw search was never the bottleneck; knowing when to abandon a branch was. The other crucial element is the release of a formal proof. Construction-type results are most vulnerable to the suspicion that the answer was simply stumbled upon by luck. A formal proof, by contrast, can be handed to a proof assistant and machine-verified line by line, squeezing the uncertainty of human review down toward a minimum. Releasing the paper and the proof together puts reproducibility squarely on the table. ## Industry Background Over the past two years, AI participation in mathematical research has been shifting from solving problems toward making discoveries. The capabilities of earlier models concentrated on competition-style questions and restatements of known theorems, which at bottom was still pattern matching. Constructing a set that nobody had found before and proving it optimal belongs to the category of creating new knowledge, and the difficulty involved is not on the same order of magnitude. The line between reciting mathematics and actually doing mathematics is precisely where the boundary has moved, and verification tooling is quietly becoming as important as the model itself. Going from more than fifty years hovering near 1.1 to arriving directly at 2 with a proof of optimality is itself telling. The way an AI system explores an extremely large search space may differ systematically from the intuitive routes human mathematicians take, and that very difference is where its value lies. ## Industry Impact For China's domestic AI industry, this is a demonstration of capability with real substance behind it. It shows that a large model combined with an agent can now reach genuine, open-ended research problems rather than just racking up points on benchmarks. The coordination among foundation model capability, agent architecture, and formal verification also offers a reference paradigm for designing research agents going forward. What deserves even closer attention is the method itself. If a research agent can produce original results in a domain as abstract as number theory, then materials science, drug discovery, and algorithm optimization, all of which likewise depend on large-scale search, are in principle open to being moved by the same class of approach. One result does not yet make a method, and the real test will be whether the same pipeline reproduces on a second open problem. Cracking a hard math problem is the outcome; the reusable research method is what actually stays behind.
